GCRIS Repository Collection: Collection of Mathematics / Matematik Bölümü koleksiyonu
Collection of Mathematics / Matematik Bölümü koleksiyonu
https://hdl.handle.net/11147/8
2024-06-19T17:13:46Z
2024-06-19T17:13:46Z
Uniform asymptotic and input to state stability by indefinite Lyapunov functions
Şahan, Gökhan
Özdemir, Durmuş
https://hdl.handle.net/11147/14270
2024-02-13T12:28:24Z
2024-01-01T00:00:00Z
Title: Uniform asymptotic and input to state stability by indefinite Lyapunov functions
Authors: Şahan, Gökhan; Özdemir, Durmuş
Abstract: In this work, we study uniform, uniform asymptotic, and input-to-state stability conditions for nonlinear time-varying systems. We introduce an easily verifiable condition for uniform attractivity by utilizing an indefinite sign upper bound for the derivative of the Lyapunov function. With this bounding structure, we propose novel conditions that enable us to test uniform stability, uniform asymptotic stability, and ISS, easily. As a result, the constraints on the coefficients of the bound that identify uniformity for stability and attractivity, and many of the available conditions have been relaxed. The results are also used for the perturbation problem of uniformly stable and uniformly asymptotically stable linear time-varying systems. Consequently, we demonstrate that uniform asymptotic stability of nonlinear time-varying systems can be robust for perturbations, but with special time-varying coefficients. © 2024 European Control Association
2024-01-01T00:00:00Z
Local well-posedness of the higher-order nonlinear Schrödinger equation on the half-line: Single-boundary condition case
Alkın, Aykut
Mantzavinos, Dionyssios
Özsarı, Türker
https://hdl.handle.net/11147/14055
2024-01-27T22:04:24Z
2023-01-01T00:00:00Z
Title: Local well-posedness of the higher-order nonlinear Schrödinger equation on the half-line: Single-boundary condition case
Authors: Alkın, Aykut; Mantzavinos, Dionyssios; Özsarı, Türker
Abstract: We establish local well-posedness in the sense of Hadamard for a certain third-order nonlinear Schrödinger equation with a multiterm linear part and a general power nonlinearity, known as higher-order nonlinear Schrödinger equation, formulated on the half-line (Formula presented.). We consider the scenario of associated coefficients such that only one boundary condition is required and hence assume a general nonhomogeneous boundary datum of Dirichlet type at (Formula presented.). Our functional framework centers around fractional Sobolev spaces (Formula presented.) with respect to the spatial variable. We treat both high regularity ((Formula presented.)) and low regularity ((Formula presented.)) solutions: in the former setting, the relevant nonlinearity can be handled via the Banach algebra property; in the latter setting, however, this is no longer the case and, instead, delicate Strichartz estimates must be established. This task is especially challenging in the framework of nonhomogeneous initial-boundary value problems, as it involves proving boundary-type Strichartz estimates that are not common in the study of Cauchy (initial value) problems. The linear analysis, which forms the core of this work, crucially relies on a weak solution formulation defined through the novel solution formulae obtained via the Fokas method (also known as the unified transform) for the associated forced linear problem. In this connection, we note that the higher-order Schrödinger equation comes with an increased level of difficulty due to the presence of more than one spatial derivatives in the linear part of the equation. This feature manifests itself via several complications throughout the analysis, including (i) analyticity issues related to complex square roots, which require careful treatment of branch cuts and deformations of integration contours; (ii) singularities that emerge upon changes of variables in the Fourier analysis arguments; and (iii) complicated oscillatory kernels in the weak solution formula for the linear initial-boundary value problem, which require a subtle analysis of the dispersion in terms of the regularity of the boundary data. The present work provides a first, complete treatment via the Fokas method of a nonhomogeneous initial-boundary value problem for a partial differential equation associated with a multiterm linear differential operator. © 2023 Wiley Periodicals LLC.
2023-01-01T00:00:00Z
On purities relative to minimal right ideals
Alagöz, Yusuf
Alizade, Rafail
Büyükaşık, Engin
Sağbaş, Selçuk
https://hdl.handle.net/11147/14028
2024-02-28T08:16:59Z
2023-01-01T00:00:00Z
Title: On purities relative to minimal right ideals
Authors: Alagöz, Yusuf; Alizade, Rafail; Büyükaşık, Engin; Sağbaş, Selçuk
Abstract: Abstract: We call a right module M weakly neat-flat if (Formula presented.) is surjective for any epimorphism (Formula presented.) and any simple right ideal S . A left module M is called weakly absolutely s-pure if (Formula presented.) is monic, for any monomorphism (Formula presented.) and any simple right ideal S . These notions are proper generalization of the neat-flat and the absolutely s-pure modules which are defined in the same way by considering all simple right modules of the ring, respectively. In this paper, we study some closure properties of weakly neat-flat and weakly absolutely s-pure modules, and investigate several classes of rings that are characterized via these modules. The relation between these modules and some well-known homological objects such as projective, flat, injective and absolutely pure are studied. For instance, it is proved that R is a right Kasch ring if and only if every weakly neat-flat right R -module is neat-flat (moreover if R is right min-coherent) if and only if every weakly absolutely s-pure left R -module is absolutely s-pure. The rings over which every weakly neat-flat (resp. weakly absolutely s-pure) module is injective and projective are exactly the QF rings. Finally, we study enveloping and covering properties of weakly neat-flat and weakly absolutely s-pure modules. The rings over which every simple right ideal has an epic projective envelope are characterized. © 2023, Pleiades Publishing, Ltd.
2023-01-01T00:00:00Z
On Schrödinger operators modified by δ interactions
Akbaş, Kaya Güven
Erman, Fatih
Turgut, O. Teoman
https://hdl.handle.net/11147/14022
2024-02-28T11:13:26Z
2023-01-01T00:00:00Z
Title: On Schrödinger operators modified by δ interactions
Authors: Akbaş, Kaya Güven; Erman, Fatih; Turgut, O. Teoman
Abstract: We study the spectral properties of a Schrödinger operator H0 modified by δ interactions and show explicitly how the poles of the new Green's function are rearranged relative to the poles of original Green's function of H0. We prove that the new bound state energies are interlaced between the old ones, and the ground state energy is always lowered if the δ interaction is attractive. We also derive an alternative perturbative method of finding the bound state energies and wave functions under the assumption of a small coupling constant in a somewhat heuristic manner. We further show that these results can be extended to cases in which a renormalization process is required. We consider the possible extensions of our results to the multi center case, to δ interaction supported on curves, and to the case, where the particle is moving in a compact two-dimensional manifold under the influence of δ interaction. Finally, the semi-relativistic extension of the last problem has been studied explicitly. © 2023 Elsevier Inc.
2023-01-01T00:00:00Z